A DECOMPOSITION THEORY FOR UNITARY ...1962] A DECOMPOSITION THEORY OF LOCALLY COMPACT GROUPS 255 in...

A DECOMPOSITION THEORY FOR UNITARYREPRESENTATIONS OF LOCALLY

COMPACT GROUPSBY

JOHN A. ERNEST(')Introduction. In the classical theory of finite dimensional representations

of compact groups, every representation may be expressed uniquely as adirect sum of irreducible representations. This reduces the problem of deter-mining the complete representation theory of such a group to the muchsimpler problem of determining the irreducible representations. In the pastdecade or so many attempts have been made to generalize this situation to atheory of (not necessarily finite dimensional) unitary representations ofseparable locally compact groups. For this purpose, the von Neumann con-cept of "direct-integrals" of weakly-closed *-algebras of operators [19] wasadapted to give a "direct-integral" of representations, which appears to bethe natural extension of the concept of a direct sum of representations [17].There is a natural "duality" between representation theory and the theory ofvon Neumann algebras. In particular, there is an intrinsic way of classifyingrepresentations into types I, II and III, which is equivalent to the Murray-von Neumann classification of the JF*-rings of operators generated by therange of the representations. A group is called type I if all its representationsare type I. (Cf. [l; 12; 14].)

In the case of type I groups, a completely satisfactory decompositiontheory is obtained. (Cf. [15] with added amendments [2; 6; 7 and 8].) Weassume throughout this paragraph that the group G is type I. The dual ob-ject is then, as in the classical case, the collection G of all unitary equivalenceclasses of irreducible representations of the group G, in which a

A DECOMPOSITION THEORY OF LOCALLY COMPACT GROUPS 253

one-to-one order preserving correspondence is set up between the collectionof all unitary equivalence classes of multiplicity free representations of a typeI group and the collection of all cr-finite measure classes on G, [15, corollary toTheorem 10.6]. The theory as developed in [15] required an additional tech-nical assumption on the Borel structure in G. The final touch was put on thetype I theory when Glimm [7] proved that this additional hypothesis (thatG be "smooth") is equivalent to the property that G be type I. (Also see [2].)

However when one leaves the frontiers of the type I case, the analogieswith the classical theory of finite dimensional representations break downrapidly. Type II and III representations have no irreducible subrepresenta-tions. Multiplicity theory (see [14]) assumes a more difficult, albeit moreinteresting, form. Representations can still be decomposed into direct inte-grals of irreducible representations, but examples were soon discovered [13,Theorem 11 ] where a type II representation may be expressed as a directintegral of irreducible representations in two different ways, so that no com-ponent of one decomposition is unitary equivalent to any component in thesecond decomposition. Further there is no hope of expressing a nontype Irepresentation as a discrete direct sum of multiplicity free representations,as every multiplicity free representation is type I. Further it has been shown[7] that the Borel structure in the dual G is necessarily bad (specifically itdoes not have a countable separating family of Borel sets) in the nontype Icase. As we shall see, all of these difficulties combine to force one to considera new "dual-object." To continue the narrative it is necessary at this pointto define some terms.

Let x—*LX and x-^Mx denote two unitary representations of a separablelocally compact group G over separable Hubert spaces 3C(L) and 3C(i7) re-spectively. L and M are said to be disjoint, denoted LàM, if no subrepre-sentation of L is equivalent to any subrepresentation of M. We say L coversM, denoted L\M, if no subrepresentation of M is disjoint from L. We sayL is quasi-equivalent to M, denoted L~M, if L covers M and M covers L.The term "quasi-equivalence" is perhaps misleading as this relation is a bona-fide equivalence relation. The adjective "quasi" is used only to distinguish therelation from the more traditional concept of unitary equivalence. As weshall see later, the collection of all quasi-equivalence classes of representa-tions, under the covering relation defined above, forms a

254 J. A. ERNEST [August

not dependent on the type I hypothesis, for distinguishing representations(up to unitary equivalence) within a quasi-equivalence class. (For examplesee [14, Theorem 1.5].) Thus the problem of classifying all representations,up to unitary equivalence, is reduced to that of classifying the quasi-equiva-lence classes. By analogy with the type I case, we plan to effect this classi-fication by means of a decomposition theory, with each quasi-equivalenceclass corresponding to a measure class on a dual-object, the points of thisdual-object representing the "building blocks."

Let Q denote the collection of all quasi-equivalence classes of representa-tions of some group G, partially ordered by the covering relation definedabove. The elements of Q which are minimal with respect to this partialordering we shall call primary classes. We define G, the quasi-dual of G, tobe the collection of these primary classes. In our general decompositiontheory, the primary classes will serve as the "building blocks" and the quasi-dual G as our "dual object." A representation will be called primary if it iscontained in a primary class. Then a representation is primary if and only ifit cannot be expressed as the direct sum of two disjoint representations. LetCt(L) denote the von Neumann algebra of operators generated by the rangeof the representation L. Then L is primary if and only if &(L) is a factor inthe Murray-von Neumann terminology, i.e., if and only if &(L) has trivialcenter. For this reason primary representations are often referred to in theliterature as "factor representations." (For example in [14 and 18].) How-ever the term "primary," used in [12] and [16], seems to be more appropriatein the context of group representations and will be used exclusively through-out this paper. By means of the Murray-von Neumann theory of factors,every primary representation, and thus every primary class, may be classifiedas being either of type I, II or III. (Cf. [12 and 14].)

A er-ring of sets, called Borel sets, is specified in G, which will hereafter becalled a "Borel structure" for G, (cf. [15]). Our main result on decompositiontheory (Theorem 2) is that the central decomposition [14, p. 201 ], often calledthe "canonical decomposition," may always be taken over G. More specifi-cally, to each representation L of G, there corresponds a measure p on Gand a ju-measurable map y—>7> such that LyEy for all yGG, L~f(¡Lvdp(y)and the range of the projection valued measure associated with this decom-position, is the collection of all projections in the center of

1962] A DECOMPOSITION THEORY OF LOCALLY COMPACT GROUPS 255

in the type I case as the canonical measure lattice is then simply the latticeof all ff-finite measure classes on G. It is hoped that additional investigationwill give more detailed knowledge of the canonical measure lattice for thenontype I case, than is given in the somewhat superficial characterization(Proposition 7) presented in this paper.

We remark at this point that although the previous statements are givenin terms of group representations, completely analogous results hold forbounded ^representations of separable Banach *-algebras, i.e., for bounded*-algebra homomorphisms into bounded linear operators on a separableHubert space. Indeed, the theory will be formulated so as to apply both tounitary representations of groups and to ""-representations of algebras.

In §1 we formulate our problem, pin down our basic definitions, and de-scribe the object, Q, which we plan to characterize. In §2 we define and de-scribe our new dual-object, the quasi-dual. In §3 we obtain some propertiesof the central decomposition which we will need in the sections to follow. In§4 we connect the two previous sections by showing that the quasi-dual mayalways be used as the Borel space in the central decomposition. We also showin this section that two representations are quasi-equivalent if and only ifthe corresponding components in their central decompositions are quasi-equivalent. In §5 we examine the lattice of measure classes determined on thequasi-dual by means of the decomposition theory developed in §4. In par-ticular we study the correspondence thus generated between the elements ofQ and measure classes on the quasi-dual. In §6 we develop a formulation ofmultiplicity theory which (a) meshes in a natural manner with the character-ization of representations up to quasi-equivalence developed in the previoussections, (b) distinguishes representations, up to unitary equivalence, withina quasi-equivalence class, and (c) casts the multiplicity theory for type I, IIand III representations into the same mold. §7 outlines the few modificationsneeded to make the entire theory applicable to projective representations.

I wish to thank Professors Feldman, Fell, Glimm and Mackey for somehelpful conversations relating to the subject matter of this paper.

Professor J. Dixmier, after reading the first draft of this work, communi-cated to the author by letter the important result (Theorem 1) that the sub-set of all primary representations Gp is a Borel subset of the set of all concreterepresentations Gc. (Cf. §2.) Lemmas 2 and 3, and Theorem 1, including theproofs, are the work of Professor Dixmier and we appreciate his willingnessto have them incorporated into this paper. Theorem 1 has improved manyof the subsequent results and in some places has greatly simplified the proofs.The reader may wish to compare this presentation with the formulation ofthe theory [4] developed before Professor Dixmier made his contribution.

1. Formulation of the problem. Throughout this paper W will denote afixed separable locally compact group or a fixed separable Banach *-algebra.When W is a group, the term "representation" shall mean a homomorphism

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


x-+Lx of *W into the group of all unitary mappings of some separable Hubertspace, 3C(£), onto itself such that x-^>Lx() is a continuous function from Wto 3C(£) for all 4> in 3C(L). When W is a Banach *-algebra, the term "repre-sentation" shall mean a homomorphism x—>Lx of VP into the algebra of allbounded linear operators on some separable Hubert space 3C(L) such that(Lx)* = LX* for all xGW, and \\LX\\ ú\\x\\ for all xGW. Two representationsL and M of W are said to be equivalent and we write £~M if there exists aunitary mapping [/ of 3C(L) onto X(M) such that ULxU~l=Mx for all a; inW. Let (R(L, L) denote the ring of all bounded linear operators on 3C(¿),which commute with every element of the range of L. L is said to be irreduci-ble if the vector space dimension of (R(L, L) is one.

Given a group G, one can construct its group algebra GLq, the set of allequivalence classes of complex valued functions on G which are integrablewith respect to left invariant Haar measure. Multiplication is defined byconvolution, i.e., [/* g](x) =Jof(xy)g(y~1)dy. We define an involution in&G by f*(x) =f(x~l)A(x~1) where A(x) is a continuous homomorphism of Ginto the real numbers satisfying fGf(xt)dx = faA(t)f(x)dx for all/GCtc andtEG. (Cf. [il, p. 120; and 15, p. 154].) Furthermore there is a one-to-onecorrespondence between the set of all representations T of G and the set ofall representations V of do which are nowhere trivial (sometimes calledproper) in the sense that the linear union of the ranges of the T'¡ for/Gßcis dense in 3C(F'). The correspondence between T and V is defined by therequirement that (T}4>, \¡/) = ./V(s)(r.c/>, ip)ds for all/G«G and d>, ypE&(T).

Lemma 1. Let T be any unitary representation of a separable locally compactgroup G and let V denote the corresponding nowhere trivial *-representation ofits group algebra aG. Then (R(F, T) = (R(T', T).

Proof. Left to the reader.From Lemma 1 it follows that the von Neumann algebra d(F) generated

by the range of the representation T of the group G is identical with GL(T'),the von Neumann algebra generated by the range of the corresponding^representation T' of the group algebra Clc?. Thus we have that T is irreduci-ble if and only if T' is irreducible. T is primary if and only if T' is primary. Tand V are of the same von Neumann type. The concept of "group C*-algebra" has been introduced in [5, p. 369]. There is an exactly similar cor-respondence between representations of the group and ^representations ofits "group C*-algebra." Lemma 1 and the statements made above also applyto this correspondence. For the most part, it is these properties of the cor-respondence between representations of a group and of its group algebra (orof its group C*-algebra) which enables one to develop a decomposition theoryfor representations of groups simultaneously with a decomposition theory for"■-representations of Banach *-algebras.

The object of investigation is the collection Q of all quasi-equivalenceclasses of representations of W. Q is partially ordered by the covering relation



defined above. In discussing this ordering it is convenient to adjoin an ele-ment, denoted 0, to Q. (One might take 0 to be the empty set.) We extend thepartial ordering by defining L) 0 for every LCQ. Then two points L, MCQ,are disjoint if and only if the infimum of L and M, L/\M, is equal to 0. Weshall continue to use the notation Q for the original collection with this dis-tinguished point adjoined.

Proposition 1. Q is a distributive a-lattice in which one may take relativecomplements. (By the "relative complement," L — M, where L, MCQ, we meanan element NCQ such that NóM and N\/M=L\/M.)

Proof. Since there is a one-to-one order preserving correspondence be-tween the Q associated with a group G and that associated with its groupalgebra &o, it is sufficient to verify the statement for the case where "W is aBanach *-algebra.

Given a countable collection of elements of Q, one need only choose onerepresentative of each class in the given collection, form their direct sum inthe ordinary representation theoretic manner, and then take the quasi-equivalence class containing the representation constructed in this way. It iseasy to verify that the element of Q determined in this manner is the supre-mum of the given countable collection of elements of Q. Hence Q is


specify a Lx in the weak operator topology and thus itis sufficient to show that ||L^||—»||La^|| for all


there exists T\, • • • , 7j in Lb(3C)f\(R(L\ L>) such that Lj,J\+ ■ ■ ■ +£'B71is contained in Zi(3C) and d(LixlT[ + ' ' ■ +-i47l «S) =a- Since L*W) isweakly compact one may, by taking a subnet of {£*}, assume that theT\, • • • , T*n converge weakly to some elements Ti, • • • , Tn in Lb(3C). Sup-pose xGW and 0, xpEX.. Then we have that (T\Ltxd>ü) = (Ltxd>, Tfy)-*(L^, Fiï) = (TiLxd>,t)and

(LXTÍ, *) = (rícfc, LÍV) -» (?>, T.!*) = (£.2-1*, *).

Hence (TiL^ü) = (LxTi,\p). Therefore FiG be two positive integers. We will complete the proof ofthe lemma by showing that there exists an « such that LEZk¡n,P. Let C(L)denote the space of finite sums of operators of the form LXT where xEW andTE&(L, L). Then C(L) is a *-algebra of operators whose weak closure is3l(L) =L(3C). According to the Kaplansky density theorem (cf. [l, p. 46])there exists an operator PEC(L) such that ||P||


d(LZ{ Pi4- • • • -\-LXi Tm, Sk) ál/p. Hence if one chooses n to be any integergreater than i\, • • • ,im and ||Pi||, • • • , ||Pm||> we will have LCZk,„,p.

Theorem 1(2). The set "W» is a Borel subset of V?c.

Proof. Give each V?n the smallest topology for which all the functionsL—=>iLx, is a complete separable metric space. It is this fact whichenables us to use the results of [10] in what follows. Form the topologicalCartesian product VX"W". Then the projections of "WX"»» into Wp arecontinuous and thus Borel. Theorem 8.2 of [15] states that #(L, M) is aBorel function on WX'W1'. (ä(L, M) is the vector-space dimension of the setof all bounded linear transformations T from 3C(L) to 3Q.(M) such thatTLX = MXT for all x£W.) Thus S={(L, M):â(L, M)^0}r\(WpXB)= {(L, M):L~M, LCWp, MCB] is a Borel subset of WX"W". By [10,p. 366] Bq, which is the projection of S into Wp, is analytic and thus, by [10,p. 391], measurable.

If B intersects each quasi-equivalence class in at most one point, then theprojection of 5 into W is one-to-one. Hence by [15, Theorem 3.2] the rangeof this projection, namely B", is Borel.

Corollary. W is a separated Borel space. Indeed, every point of V? is aBorel set.



Proof. If L is a point of W, choose LEW such that LEL. {L} is a Borelset in V?p and hence by Proposition 2, {L} « is Borel. Thus {L} is a Borelset in *W.

3. The central decomposition. We have described our dual object W andwe must now develop a method of associating a measure p on W with eachrepresentation L of W. This will be effected by the central decomposition ofL (sometimes called the "canonical decomposition"). This section will bedevoted to the description of this concept. (Cf. [14; 15; 17; 19 and 20].)

Recall that a Borel space S is called standard if it is Borel isomorphicwith the Borel space associated with a complete separable metric space. Ameasure p on a Borel space 5 is called standard if 5 contains a Borel set 73such that p(S—B) = 0 and 73 is a standard Borel space. For completeness wegive the definition of direct integral representation as formulated by G. W.Mackey in [15]. Let p denote a cr-finite standard measure in the Borel space5 and suppose y—>7> is a ju-measurable mapping of 5 into W. For eachre= », 1, 2, • • • , let 3C„ denote the classical Hubert space of «-tuples ofcomplex numbers referred to in the definition of "W0 above. Let Sn denote thesubset of 5 consisting of those y ES for which the representation space of7>, 3C(7>), is 3C„. Form the Hubert space 5Q.(M) consisting of all functions/from 5 to UB-«o,i,»,• • • 3Cn such that (a) /(y)G3Cn for all yESn; (b) for eachre= », 1, 2, • • • , (f(y), G3C„;(c) fs(f(y), f(y))dp(y) < ». Define an inner product in 3C(M) by setting(/■ g) =fs(f(y), g(y))dp(y) for/, gEK(M) and identify/ and g if ||/-g|| =0.For each xGW and /G3C(M), let Mx(f)=g where g(y)=Lvx(f(y)) for ally ES. Then the map x-+Mx is a representation of W which we define to bethe direct integral of y—>Lv and we write M = fsLydp(y).

To each Borel subset 73 of 5 we associate a projection £(73) on 5C(M) de-fined by (E(B)f)(y)=x(B)f(y) where x(-B) = 1 when yG73 and X(B) = 0 wheny EB. The function 73—»£(73) is called a projection valued measure. The de-composition M = fsLvdp(y) is called central (or canonical) when the range ofthe corresponding projection valued measure is just the set of all projectionscontained in the center of (R(M, M). Every representation L of V? has a cen-tral decomposition. Further almost all the components of the central decom-position are primary [14, Theorem 2.5]. The central decomposition has thecharacterizing property that any decomposition of a representation L intoprimary representations over a standard Borel space is necessarily a "refine-ment," in the obvious sense, of the central decomposition of L. Further thecentral decomposition of a representation is unique in the sense that ifL~fs>Lv'dp'(y') is another central decomposition for L, then there exists ameasure space isomorphism between (S, p) and (S', p'), say y—*y' such that7>~7>' for /¿-almost all y in S.

A. Guichardet [8] has proved that if a multiplicity-free representationis decomposed into irreducible representations, then the components are,almost everywhere, two-by-two disjoint. A perturbation of his proof gives



the following more general result (Proposition 3). We need a proof directlyapplicable to groups in order to obtain an easy generalization to projectiverepresentations, which will be discussed in §7. For this reason we present therevised proof for the case where W is a separable locally compact group. Therequired revision of Guichardet's proof in the case where W is a *-algebra isleft to the interested reader.

Proposition 3. Let L denote a representation of V? with a decompositionLcafsLydp(y) over a separable Borel space S, such that the range (S> of the cor-responding projection-valued measure is contained in the center, Q(R(L), of thecommuting algebra , Lv') suchthat PvÉnRiy, y') = Riy, y')P\. However each P|n is either 0 or the identityoperator. Hence the last equality implies thatP\ = P\n. Let S' = Dn„i,2,... Sn.Then ju(5—5')= ^„/x(S-S„) = 0. Now suppose y, y'CS' and that 7> andL"' are not disjoint. Then P|n = 7£n for »=1, 2, • • • , which implies thaty'££„ if and only if yCEn. Since the collection {En} separates points wehave y = y'.

Corollary. The central decomposition of a representation has the propertythat, after eliminating a set of measure zero, the components are two-by-two dis-joint primary representations^).

(*) This result has been announced by M. A. Naimark [18, Theorem l].



4. Decomposition theory. In this section we connect the two previoussections by proving that the Borel space 5 associated with the central decom-position of a representation of W may always be taken to be V?. We then con-sider the correspondence thus induced between representations of W andmeasures on W.

Theorem 2. The central decomposition of a representation of V? may alwaysbe taken over the space W. More explicitly, to every representation L of W therecorresponds a standard o-finite Borel measure p on W and a p-measurable mapy-+Ly of W into W such that LyEy, Lc^.f^Lvdp(y), and the range of the pro-jection valued measure associated with this decomposition is the set of all projec-tions in e(R(L).

Proof. Let Lc^f sLvdp(y) be the central decomposition of L over somestandard Borel space S, where y—>7> is a Borel map of 5 into V?". By thecorollary to Proposition 3 we may assume that the components 7> are two-by-two disjoint primary representations. Thus the map y—»7> is a one-to-oneBorel map of 5 into Wp, which, by [15, Theorem 3.2] is a Borel isomorphismof 5 onto a Borel subset S' of W. Let

of B into Wc such thatLvEy for all yEB and fBL"dp(y) is the central decomposition of L.

The remainder of this section will be devoted to showing that there existsa one-to-one correspondence between elements of Q (i.e., quasi-equivalenceclasses of representations) and certain standard measure classes on W. Wedo this by proving that two representations of V? are quasi-equivalent if andonly if they correspond, according to the corollary to Theorem 2, to the samestandard measure class on V?. In order to obtain this result, however, wemust first digress to clarify the relationship between the decomposition



theory of representations and the well-known decomposition theory forvon Neumann algebras.

Suppose Lc^JsLydp(y) is a direct integral decomposition (not necessarilycentral) of a representation L of "W. Let CL(L) and 0(7,") denote the von Neu-mann algebras generated by the range of L and the range of Ly respectively.Then it is always true that y—>d(L!/) is a measurable field of von Neumannalgebras and furthermore 6,(L)Cfsd(Ly)dp(y). (See [l] for terminology andthe theory of direct integrals of von Neumann algebras.) Indeed, by choosinga dense sequence of elements of °W (we are assuming throughout that W isseparable), say Xi, x2, • • ■ , we obtain a sequence of operator fields, y—»7,^,y—>L"Xi, • • • , such that, for ju-almost all y, {Lyx.} generates ft(7>). FurtherJsQ(Ly)dp(y) is a von Neumann algebra which contains the decomposableoperators LXi = fsLx.dp(y) for all *, and thus fs&(Ly)dp(y) contains ß(L).Note, however, that &(L) may well be distinct from fs&(Ly)dp(y). In factthis is always the case when the Boolean algebra of projections associatedwith the decomposition is not contained in the center of (R(L, L). Indeed, sup-pose £ is in the range of the associated projection valued measure and££C(ft(7,), the center of (R(L, L). In the terminology of Dixmier [l] £ is adiagonalizable projection and thus [l, Theorem 1, p. 178] ECfs&iLy)dpiy).However, since ££(R(L, L) and ££C(n(L) = (R(L, L)f\&(L), we have££a(7,). The exact situation is described in the following statement.

Proposition 4. Suppose L = JsLvdp(y). A necessary and sufficient conditionfor &(L) to be equal to fs&(Ly)dp(y) is that the Boolean algebra of projections 03,associated with the given decomposition of L, be contained in the center of &(L).

Proof. The necessity of the condition was given in the previous remark.Suppose now the condition holds. Letting {xi, x2, • • • } denote a countabledense collection of elements of W we have [l, Theorem 1, p. 178] thatJs&(Ly)dp(y) is generated by {fsLx.dp(y)=LXi:i=l, 2, • • • } and the col-lection of diagonalizable operators. But the collection of diagonalizable oper-ators is just the von Neumann algebra generated by 03, which is containedin d(L). Thus fs&(Ly)dp(y) £Ct(L). 0i(L), as we remarked above, is alwayscontained in fs&(Ly)dp(y).

Corollary 1. If L is a representation of W with central decompositionL=J^çLydp(y), then a(L)=J^a(Ly)dp(y).

Corollary 2. If L is a representation of \V with a decompositionL = JsLydp(y) such that the associated Boolean algebra of projections is con-tained in the center of &(L), then

(R(L, L)= f &(Ly, Ly)dp(y)J s

and



Q(R(L) = j e&(L»)dp(y).J s

In particular these properties hold for the central decomposition.

Proof. By Proposition 4 we have a(L)=fsQ,(Lv)dp(y). Then by [l,Theorem 4, p. 184] we have


*-representations of the group algebra G,a. Then L~M if and only if L'~M'.Indeed let


Proposition 5. Suppose L and M are representations of W with centraldecompositions over the same measure space, say L = JsL"dp(y) andM=fsM"dp(y). If Lv~M"for p-almost all y, then L~M.

Proof. We first restrict ourselves to the special case where W is a Banach*-algebra and L and M are nowhere trivial ""-representations of W. ByLemma 4 we have, for /¿-almost all y in S, that there exists an algebraic iso-morphism ) onto a(M") such that r"G Ct(7>) is a measurable field of operators, then y-*py(Tv)E&(My) is a measurable field of operators. If y-+TvE&(Lv) is a measurablefield of operators, we may form T=fsTvdp(y) and by Proposition 4, TE&(L).By Lemma 7 there exists a sequence of elements, say LXi in the range of Lsuch that LXi converges to T is the strong topology and furthermore ||LiJ|is || r|| for all i. Further each LXi is decomposable and therefore we may writeLXi = fsLvx.dp(y), where, for each i, y^>LvXi is a /¿-measurable field of operators.Further, by [l, Proposition 4(i), p. 162], there exists a subsequence LXih ofLXi such that Lx. converges to 7" in the strong topology and further, for p-almost all y, L\\ converges to T" in the strong topology. Thus we may as-sume, without loss of generality, that we have a sequence LXi such that LXiconverges to T and Vx. converges to T" for /¿-almost all y, in the strong topol-ogy. Now for each i, MvXi is a measura-ble field of operators. By [l, Corollary 1, p. 57],


of V?. Then L and M have central decompositions over the same measure space,L=JsLvdp(y) and M = fsMydpiy) such that Ly~My for p-almost all y.

Proof. Let L = JsLydpiy) denote the central decomposition of L. Let


Theorem 4. The one-to-one correspondence between the elements of Q andelements of 9TC(cvv') given by Theorems 2 and 3 is order preserving. That is tosay, if L, MEQ, then L}M if and only if e(L) g£ Q(M), where e(L) and Q(M)denote the measure classes on V? corresponding to L and M respectively.

Proof. Suppose first that L covers M. Then it is well known (see Proposi-tion 1 and its proof) that L may be written L^¿Lí-\-L2 where Ll~M andL2àM and LX6L2. Suppose Lc^f^Lvdß(y), where p=Q(L). Then sinceLl6L2, there exists a projection £ in C(R(L) such that Ll = LE and L2= £(/ — £). Since the above decomposition is central, £ is in the range of thecorresponding projection valued measure. Thus there exists a Borel subset73 CW such that Ll~fBLvdfi(y) and L2c^.f^_B)Lvdß(y). Define the measureir on W by ir(C) = ß(C(~\B) for all Borel sets C in W. Then L1 has central de-composition Llc^.f^Lvd-K(y). Clearly ß^ir. However, since LL~M we have,by Theorem 3, that iro^v where v= Q(M). Hence p^v.

Conversely, suppose /¿ = y» where p—G(L) and v=Q(M). Then we maywrite p = pi\/pi where /ti—>>, pi-Lv and /¿i-L/¿2. Indeed, there exists a Borel set73 CW such that ßi(C) = ß(CC\B) and ßi(C) = ß(Cr\(V?-B)) for all Borelsets CCW. Cf. [9, Theorem 2, p. 78]. Then L~fMif and only if Q(L) _Le(Af).

We next point out that the canonical lattice ideal is, in general, a properideal of the lattice of all standard measures on W. Indeed, consider the exam-ple given in [13, pp. 590-591] of a group V? which is the semi-direct product



of the additive group of rational numbers Gi with the multiplicative group ofnonzero rational numbers G2. Then the right regular representation R of V?,which is a type II primary representation, can be expressed as a directintegral of irreducible representations, R~f§2Lydp(y) where G2 is the charac-ter group of G2 and p is Haar measure on G2. By [15, corollary to Theorem8.7], G2 is smooth and hence by [7, Theorem 2], G2 is a standard Borel space.Further the components in this decomposition are two-by-two disjoint. Thuswe may proceed, by the method used in the proof of Theorem 2, to obtain astandard tr-finite Borel measure p on W such that R~f


Proof. Suppose that p is contained in GSKiW). Then there exists a /x-measurable map y—>7> of "W into W such that 7>£y and let p' be the cor-responding Q-valued measure on W. Let Pi, B2 denote Borel subsets of "Wsuch that Pi £ B2. Then Pi H (P2 - Pi) is empty and thus JBlLydpiy)6 f(B,-Bl)Lydpiy) which implies that /¿'(Pi) 6 p'iB2 — Pi). Since /¿'(P2)= /¿'(PAJ(P2-Pi)) =/¿'(Pi) Vp'iB2-Bi), we have, by the definition of "rela-tive complement," that /¿'(P2-P1) =ßiB2) -/¿'(Pi).

Conversely, suppose /¿ is a standard measure class on V?, that y—>7," is a/¿-measurable map of W into W such that LyCy and that the correspondingQ-valued measure /¿' is subtractive. Let B denote a Borel subset of V?. Then/i'(cW-P)=/¿'(eW)-/¿'(P) and thus p'(W-B) bp'(B). Hence Jc^-B)Lydp(y)ófBLydp(y) for every Borel set P£W. Hence, by [14, Lemma 1.1 ], the range

of the projection valued measure associated with the decomposition J^Lydp(y)is contained in the center 661(7,) of the commuting algebra,


=/(^i) ; and (c) if a measure class v is in the domain of / and is the supremumof a countable family {j'y} of two-by-two disjoint, nonzero measure classeson W, then/(0=inf{/(l'y)}.

We next proceed to define a multiplicity function for p, corresponding toeach representation LEL. [14, Theorem 1.2] states that L may be expresseduniquely in the form L = LiV£iiV-Liii such that Z,¿ó£y if i^j, and L,- iseither empty or LiEQ and is of type i, for i, j = l, II, III. Because the cor-respondence between Q and e3TC('W) is lattice preserving it follows that pmay be decomposed uniquely in the form /¿ = /¿iVmiiV/¿iii such that /¿¿= Q(Li)and pi-Lpj Hi 9¿j, i,j = l, II, III. We call /¿¿the type i part of p, for«=I, II, III.

Each LEL then has a decomposition of the form Lc^f^L'dp(y)= 2Î-I J*


Proposition 8. Suppose LCQ, MCQ, LCL, MCM, /¿=6(L), andv= QiM). Let L = J^Lydpiy) and M = f^Mvdv{y) represent the central decom-positions of L and M respectively. Then L^M if and only if p^v and Ly^Myfor p-almost all y.

Proof. We remark first that in this proof and the material to follow, weadopt the usual practice of identifying representations which are unitarilyequivalent, whenever it is convenient to do so.

Suppose first that L^M. Then L\M and v — p\/p' where /¿-L/¿'. LetMl = J,for /¿-almost all y. According to [14, Corollary 1 to Theorem 2.8], y^>n{y)is a /¿-measurable function. We next prove the analogue of this result for thetype II case.

Proposition 9. Let L denote a type II representation with central decomposi-tion L = f'fyLydpiy). Let M denote a finite type II representation quasi-equiva-lent to L, with central decomposition M=J%j%Mydpiy). Then Ly = miy)My forp-almost all y, where, for each y, miy) is a positive real number or oo. The func-tion y-^miy) is p-measurable.

Proof. Given anyX>0, we may form the representation M\ = f^\Mvdpiy).Further L and M\ may each be decomposed into disjoint parts, say 7, = L14-P2and MX = M{ + Ml, such that L^M{ and L2^M\. (Cf. [14, p. 198] or [12,Theorem 1.13, p. 24].) These decompositions are effected by a pair of projec-tions £ and F such that ££6(R(P) and P£6tft(M). Since L~M\ it followsthat Ll~M\ and 7,2~Afx and thus that £ and F correspond to the sameBorel set, say B of W. Hence by Proposition 8 we have Ly = miy)My-£\My for/¿-almost all y in B and Ly = miy) My¡t\My for /¿-almost all y in (W—B).

Let JX¿} denote an increasing sequence of positive real numbers whichapproaches X from the left. Then for each i, there exists a measurable set 3TC,-such that miy) ^X¿ for all y in 3R,-, and a measurable set 3R/ such thatw(y)^X,-for all y in 3H/, and ¿¿(^-(311^311/)) =0. Let 3H = U¿3rt,-. If y£3TC,



then m(y)^\i for some i and thus m(y)


Corollary 1. Assume the hypothesis of Theorem 5. Then LxcaL2 if and onlytf fn.u(L\, v) =/„,m(7,2, v) for all v^p.

Corollary 2. Let T-i and L2 denote any two representations of V?, Lx and L2the quasi-equivalence classes containing Li and L2 respectively, and /¿i=6(7,i)and /¿2=6(L2). Let M2 denote a finite representation contained in the type IIpart of L2 and let Mi denote the subrepresentation of M2 having the property thatLi} Mi and (72—Pi) à Mi. Then Li^L2 if and only if pi^p2 and f^.M^Li, v)Si/M.if,(I». v) for allv^pi.

We conclude this section with a few remarks on the type I case. For atype I quasi-equivalence class L the multiplicity function defined above isnot "relative" and we may suppress the subscript Min the notation/^(T,, v).Then we have that a representation L is multiplicity free if and only if/„(L, v) = 1 for all v^p, v^O. A type I representation L is said to be uniformlyof multiplicity n if L = nM, where M is a multiplicity free representation. (Cf.[12, p. 4l].) A measure p is said to be uniformly of multiplicity n relative tosome multiplicity function / which has p in its domain if/(c) —n for all vt^Oand v^p. (Cf. [9, p. 81 ].) These two concepts are then related in the followingmanner.

Let L denote a type I quasi-equivalence class, p = Q(L), and 7,£7,. ThenL is uniformly of multiplicity n if and only if p is uniformly of multiplicity« relative to the multiplicity function/^(L, •)•

We remark here that Theorem 1.4 of [14] may now be obtained as an easyconsequence of well-known properties of multiplicity functions. Indeed, let Ldenote any type I representation. By [9, Theorem 3, p. 82], /¿= V//¿, where\Pj\ is a countable orthogonal family of measure classes such that each /¿;either has uniform multiplicity/, relative to the multiplicity function f„(L, •),or/¿/ = 0, for/= oo, 1, 2, • • • . Hence L may be decomposed L = ^,j Lj wherethe components are two-by-two disjoint and L¡ is uniformly of multiplicity/.(Some terms may not appear.) But this is just [14, Theorem 1.4], except thatthe uniqueness of the decomposition has not been proven here.

7. Projective representations. We now restrict ourselves to the case whereW is a separable locally compact group. In certain branches of physics andin the study of induced representations it has been found necessary to enlargethe scope of representation theory to include a more general concept, that of"projective representation." A projective representation Lola separable locallycompact group G is a mapping x—»7,x of G into the group of all unitary oper-ators on a separable Hilbert space 3C(P) such that (a) the image of the identityelement e of G is the identity operator 7, (b) for all x and y in G, Lxy=


decomposition theory which is not restricted to type I representations isgreatly enhanced in this more general situation since there exist commutativegroups which have primary cr-representations which are not type I. Furtherthere exists a commutative group G and a multiplier a such that G has justone irreducible c-representation and that one is infinite dimensional. Cf.[16]. We now consider those modifications in the theory developed in theprevious sections, which are needed to obtain, for a fixed multiplier a, ageneral decomposition theory for cr-representations of G.

Let Gc'° denote the collection of concrete cr-representations which have aclassical Hubert space of re-tuples of complex numbers for a representationspace. We give Gc,ff a Borel structure exactly the way we gave W a Borelstructure. G" then denotes the collection of unitary equivalence classes ofirreducible cr-representations of G and we give G" a Borel structure in thesame manner as for G. We remark here that it has been shown in [16, §3]that Theorems 8.1 through 8.6 of [15] remain true when ftc and ft are re-placed by Gc'" and G" respectively.

Let Gp


Neumann algebra generated by the range of L. Suppose P£ft(P) and»i = ||r||. By Kaplansky's density theorem [l, Theorem 3, p. 46], the ball(Xm of Ö, of radius m, is everywhere dense, in the strong topology, in Ctm, theball of Q,(L) of radius m. Thus P is contained in the strong closure of Qm. By[l, corollary to Proposition 1, p. 33], there exists a sequence of elements indm which converge strongly to P.

We leave it to the reader to make the necessary modifications in the proofof Proposition 5, using Lemma 7' in place of 7, to make it apply to

A DECOMPOSITION THEORY FOR UNITARY ...1962] A DECOMPOSITION THEORY OF LOCALLY COMPACT GROUPS 255 in...

Documents

Transcript of A DECOMPOSITION THEORY FOR UNITARY ...1962] A DECOMPOSITION THEORY OF LOCALLY COMPACT GROUPS 255 in...